Chilean Lake District in Mid-May? Advice Needed!

I'm heading to Chile for a 3 week trip mid may to early June and I have a full week open to explore an area. I was wondering if the lake district would be worth visiting in mid May? I know this is late Fall/start of Winter but it looks like there's a lot to explore even with cooler weather and I'd love to experience some of the more Patagonia-style nature that Chile has to offer. Any thoughts on if it's worth exploring during this time? Will hiking be an option? Any recommendations on where to go in the Lakes District? I'm also open to suggestions on mountainous areas closer to Santiago for weekend trips!

I am already planning to go to the Atacama Desert, Santiago, and Easter Island so just trying to see if I could add this region to my trip, especially with a full week free!